It starts quite simple but great opportunities for number discoveries and patterns!

Investigating Pascal's Triangle

Age 7 to 14 Challenge Level:

I think that it's time to look at Pascal's Triangle afresh. So,
let's see what happens when we turn it around in a special way.
So we start with the layout as usual, turn it anticlockwise 45
degrees and then take off the top line of ones.
So the first 9 lines would look like this ;

BUT if we take the digital roots of the numbers and see the
first 30 lines we have a new look to Pascal's Triangle as shown
below.

So here you have an awful lot of numbers in rows and columns to
explore.

Those of you with some spreadsheet experience (or those of you that
are quick to learn such things!) should go to the
HINT at this point, before going further.

Some starting points for further explorations.

1. On my spreadsheet I decided to explore the rows that are
multiples of 9 as I had noticed that there were a lot of 141's and
171's in those lines. Because that is what I wanted to explore I
only wrote down the first number that comes after the patterns I
was exploring [that's why there are lots of spaces in two out of
every three rows.]

So you could do a similar thing, looking at line numbers of a
different multiple.
If you do a few you then could compare your discoveries.

2. You could take just one line [e.g.30] and use the spreadsheet to
extend it very far.
Here I've taken line 30, and grouped the numbers in that row in 9's
and then three 9's together and underneath each other. I did this
because my eyes led me to some interesting repetitions. I then
noted down things according to the columns I had placed them
under.

See what other things you can notice with this line or any other
that you choose.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice.